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817 y Fn(The)21 b(idea)h(of)f (Elkies)i(is)f(to)f(w)o(ork)f(with)i(p)q(olynomials)g Fk(g)1070 824 y Ff(`)1086 817 y Fn(\()p Fk(X)t Fn(\))e(with)i(lo)o(w)o(er)f(degree)g (than)g(the)h Fk(`)p Fn(-division)0 871 y(p)q(olynomials)17 b Fk(f)275 878 y Ff(`)291 871 y Fn(.)50 963 y Fm(4.1.)25 b(Subsets)18 b(of)f(p)q(oin)o(ts)i(of)e Fk(`)p Fm(-division.)24 b Fn(W)l(e)16 b(recall)g(that)f Fk(E)i Fn(is)f(isomorphic)h(to)d Fc(C)7 b Fk(=)p Fc(L)p Fn(\()p Fk(!)1687 970 y Fj(1)1707 963 y Fk(;)h(!)1756 970 y Fj(2)1774 963 y Fn(\).)20 b(Let)c(us)0 1017 y(consider)g Fk(x)203 1024 y Ff(r)234 1017 y Fn(=)d Fk(})p Fn(\()p Fk(r)q(!)379 1024 y Fj(1)397 1017 y Fk(=`)p Fn(\),)i(for)f(1)e Fe(\024)h Fk(r)h Fe(\024)f Fk(d)f Fn(=)h(\()p Fk(`)d Fe(\000)g Fn(1\))p Fk(=)p Fn(2.)19 b(Let)c Fk(p)1119 1024 y Fj(1)1153 1017 y Fn(b)q(e)h(the)f (sum)g Fk(p)1412 1024 y Fj(1)1443 1017 y Fn(=)e Fk(x)1517 1024 y Fj(1)1546 1017 y Fn(+)e Fe(\001)d(\001)g(\001)g Fn(+)i Fk(x)1726 1024 y Ff(d)1746 1017 y Fn(.)50 1108 y 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eop %%Page: 132 4 132 3 bop 817 141 a Fm(5.)25 b(Conclusion)50 220 y Fn(The)17 b(implemen)o(tation)h(is)f(no)o(w)f(using)h(fast)f(p)q(olynomial)i (arithmetic,)f(whic)o(h)h(has)e(pro)o(v)o(en)g(to)g(b)q(e)i(sup)q(erior.)0 274 y(The)d(op)q(erations)h(includes)h(fast)e(m)o(ultiplication)i(\(with)f (FFT,)d(done)j(b)o(y)f(R.)g(Lercier\),)h(fast)f(division,)h(gcd,)f Fk(:)8 b(:)g(:)50 328 y Fn(Mo)q(dular)16 b(equations)g(for)f Fk(`)f Fe(\024)g Fn(500)h(can)h(b)q(e)g(found,)g(with)h(Chinese)g(remainder)f (theorem)g(on)f(64)h(bits.)22 b(Tw)o(o)0 382 y(curv)o(es)15 b(are)g(studied)h(as)f(examples:)102 448 y Fm({)21 b Fn(the)15 b Fi(INRIA)h Fn(curv)o(e:)k Fk(y)552 432 y Fj(2)583 448 y Fn(=)13 b Fk(x)657 432 y Fj(3)686 448 y Fn(+)d(105)p Fk(x)g Fn(+)g(78153;)102 502 y Fm({)21 b Fn(the)15 b Fi(Pol)m(ytechnique)f Fn(curv)o(e:)20 b Fk(y)746 486 y Fj(2)778 502 y Fn(=)13 b Fk(x)852 486 y Fj(3)880 502 y Fn(+)e(4589)p Fk(x)e Fn(+)h(91128.)50 569 y(Morain's)i(record)g(is)h Fi(Pol)m(ytechnique)f 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b(p)q(oin)o(ts)g(on)g(elliptic)g(curv)o (es)i(o)o(v)o(er)e(\014nite)h(\014elds.)f({)26 b(F)m(ebruary)17 b Fa(1994)p Fr(.)e(Submitted)g(for)65 1034 y(publication)d(of)i(the)g(Actes)i (des)e(Journ)o(\023)-20 b(ees)16 b(Arithm)o(\023)-20 b(etiques)13 b(1993.)944 2749 y(132)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF